The martingale concentration in Theorem  matches the matrix Bernstein results for independent case [1]}. For any \(\delta \in (0, 1)\) , if we let the right hand side of (REF ) be \(\delta /(t+1)^2\) , then we obtain that, with probability at least \(1 - \delta /(t+1)^2\) ,
\({\bar{}_t} \le 8\Upsilon _E\sqrt{\log {\frac{d(t+1)}{\delta }}}{\sqrt{\sum _{i=0}^{t}z_{i,t}^2} \; \vee \; \sqrt{\log {\frac{d(t+1)}{\delta }}} \cdot z_t^{(\max )}}.\) 
